Rhythmic neuromorphic control of a pendulum: A hybrid systems analysis

Neuromorphic engineering is an emerging research domain that aims to realize important implementation advantages that brain-inspired technologies can offer over classical digital technologies, including energy efficiency, adaptability, and robustness. For the field of systems and control, neuromorphic controllers could potentially bring many benefits, but their advancement is hampered by lack of systematic analysis and design tools. In this paper, the objective is to show that hybrid systems methods can aid in filling this gap. We do this by formally analyzing rhythmic neuromorphic control of a pendulum system, which was recently proposed as a prototypical setup. The neuromorphic controller generates spikes, which we model as a Dirac delta pulse, whenever the pendulum angular position crosses its resting position, with the goal of inducing a stable limit cycle. This leads to modeling the closed-loop system as a hybrid dynamical system, which in between spikes evolves in open loop and where the jumps correspond to the spiking control actions. Exploiting the hybrid system model, we formally prove the existence, uniqueness, and a stability property of the hybrid limit cycle for the closed-loop system. Numerical simulations illustrate our approach. We finally elaborate on a possible spiking adaptation mechanism on the pulse amplitude to generate a hybrid limit cycle of a desired maximal angular amplitude.